Evaluate 𝑙𝑖𝑚𝜃→0 [𝑠𝑖𝑛𝜃−𝑡𝑎𝑛𝜃𝑠𝑖𝑛3 𝜃 ]

To evaluate the limit as 𝜃 approaches 0 of [𝑠𝑖𝑛𝜃−𝑡𝑎𝑛𝜃𝑠𝑖𝑛3 𝜃], we can use the properties of limits and trigonometric identities.

Step 1: Simplify the expression inside the limit: 𝑠𝑖𝑛𝜃−𝑡𝑎𝑛𝜃𝑠𝑖𝑛3 𝜃 = 𝑠𝑖𝑛𝜃 − 𝑠𝑖𝑛3 𝜃/𝑐𝑜𝑠𝜃

Step 2: Apply the trigonometric identity sin(𝑎) − sin(𝑏) = 2sin((𝑎−𝑏)/2)cos((𝑎+𝑏)/2) to simplify further: 𝑠𝑖𝑛𝜃 − 𝑠𝑖𝑛3 𝜃/𝑐𝑜𝑠𝜃 = 2sin((𝜃−3𝜃)/2)cos((𝜃+3𝜃)/2)/𝑐𝑜𝑠𝜃

Simplifying the angles: 2sin((-2𝜃)/2)cos((4𝜃)/2)/𝑐𝑜𝑠𝜃 = 2sin(-𝜃)cos(2𝜃)/𝑐𝑜𝑠𝜃

Step 3: Apply the trigonometric identity sin(-𝑎) = -sin(𝑎) to simplify further: 2sin(-𝜃)cos(2𝜃)/𝑐𝑜𝑠𝜃 = -2sin(𝜃)cos(2𝜃)/𝑐𝑜𝑠𝜃

Step 4: Apply the trigonometric identity cos(2𝑎) = cos^2(𝑎) - sin^2(𝑎) to simplify further: -2sin(𝜃)cos(2𝜃)/𝑐𝑜𝑠𝜃 = -2sin(𝜃)(cos^2(𝜃) - sin^2(𝜃))/𝑐𝑜𝑠𝜃

Step 5: Apply the trigonometric identity sin^2(𝑎) = 1 - cos^2(𝑎) to simplify further: -2sin(𝜃)(cos^2(𝜃) - sin^2(𝜃))/𝑐𝑜𝑠𝜃 = -2sin(𝜃)(1 - cos^2(𝜃) - sin^2(𝜃))/𝑐𝑜𝑠𝜃

Step 6: Simplify the expression further: -2sin(𝜃)(1 - (1 - sin^2(𝜃)) - sin^2(𝜃))/𝑐𝑜𝑠𝜃 = -2sin(𝜃)(1 - 1 + sin^2(𝜃) - sin^2(𝜃))/𝑐𝑜𝑠𝜃

Step 7: Simplify the expression further: -2sin(𝜃)(2sin^2(𝜃))/𝑐𝑜𝑠𝜃 = -4sin^3(𝜃)/𝑐𝑜𝑠𝜃

Step 8: Apply the trigonometric identity sin(𝑎)/cos(𝑎) = tan(𝑎) to simplify further: -4sin^3(𝜃)/𝑐𝑜𝑠𝜃 = -4sin^3(𝜃)/sin(𝜃)/cos(𝜃)

Step 9: Simplify the expression further: -4sin^2(𝜃)/cos(𝜃) = -4sin(𝜃)/cos(𝜃)

Step 10: Apply the trigonometric identity sin(𝑎)/cos(𝑎) = tan(𝑎) to simplify further: -4sin(𝜃)/cos(𝜃) = -4tan(𝜃)

Therefore, the limit as 𝜃 approaches 0 of [𝑠𝑖𝑛𝜃−𝑡𝑎𝑛𝜃𝑠𝑖𝑛3 𝜃] is equal to -4tan(𝜃).